The final version of the mean value theorem for harmonic functions
Delsarte's two-circle theorem states that a locally integrable function \(f\) in \(\mathbb{R}^n\), \(n\geq 2\), is harmonic if it has the mean-value property on two spheres with radii \(r_1\) and \(r_2\) where \({r_1\over r_2}\) is not a ratio of roots of a special entire function \(n(z)\). A local version of the two-circle theorem [\textit{C. A. Berenstein} and \textit{R. Gay}, Isr. J. Math. 55, 267-388 (1986; Zbl 0624.31002)] considers the case when \(f\) is defined in a ball of radius \(R>r_1+r_2\). In this paper, the author investigates the case \(R\leq r_1+r_2\) to obtain the definitive version of the two-circle theorem.
scientific article; zbMATH DE number 1028516
scientific article